If the stock price is below the exercise price, the call is worthless, but the bond matures to
C H A P T E R
2 0
Options Markets:
Introduction
699
exceeds
X, then the payoff to the call,
S
T
2 X, is added to the face value of the bond to
provide a total payoff of S
T
. The payoff to this portfolio is precisely identical to the payoff
of the protective put that we derived in Table 20.1 .
If two portfolios always provide equal values, then they must cost the same amount to
establish. Therefore, the call-plus-bond portfolio must cost the same as the stock-plus-put
portfolio. Each call costs C. The riskless zero-coupon bond costs X /(1 1 r
f
)
T
. Therefore,
the call-plus-bond portfolio costs C 1 X /(1 1 r
f
)
T
. The stock costs
S
0
to purchase now (at
time zero), while the put costs
P. Therefore, we conclude that
C
1
X
(1
1 r
f
)
T
5 S
0
1 P
(20.1)
Equation 20.1 is called the put-call parity theorem because it represents the proper
relationship between put and call prices. If the parity relation is ever violated, an arbi-
trage opportunity arises. For example, suppose you collect these data for a certain
stock:
Stock price
$110
Call price (1-year expiration, X 5 $105)
$ 17
Put price (1-year expiration, X 5 $105)
$ 5
Risk-free interest rate
5%
per year
We can use these data in Equation 20.1 to see if parity is violated:
C
1
X
(1
1 r
f
)
T
5
?
S
0
1 P
17
1
105
1.05
5
?
110
1 5
117
2 115
This result, a violation of parity—117 does not equal 115—indicates mispricing. To exploit
the mispricing, you buy the relatively cheap portfolio (the stock-plus-put position repre-
sented on the right-hand side of the equation) and sell the relatively expensive portfolio
(the call-plus-bond position corresponding to the left-hand side). Therefore, if you buy the
stock, buy the put, write the call, and borrow $100 for 1 year (because borrowing money is
the opposite of buying a bond), you should earn arbitrage profits.
Let’s examine the payoff to this strategy. In 1 year, the stock will be worth S
T
. The $100
borrowed will be paid back with interest, resulting in a cash outflow of $105. The written
call will result in a cash outflow of S
T
2 $105 if S
T
exceeds $105. The purchased put pays
off $105 2 S
T
if the stock price is below $105.
Table 20.5 summarizes the outcome. The immediate cash inflow is $2. In 1 year, the
various positions provide exactly offsetting cash flows: The $2 inflow is realized with-
out any offsetting outflows. This is an arbitrage opportunity that investors will pursue on
a large scale until buying and selling pressure restores the parity condition expressed in
Equation 20.1.
Equation 20.1 actually applies only to options on stocks that pay no dividends before
the expiration date of the option. The extension of the parity condition for European call
options on dividend-paying stocks is, however, straightforward. Problem 12 at the end of
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700
P A R T V I
Options, Futures, and Other Derivatives
the chapter leads you through the demonstration. The more general formulation of the
put-call parity condition is
P
5
C 2
S
0
1 PV(X) 1 PV(dividends)
(20.2)
where PV(dividends) is the present value of the dividends that will be paid by the stock
during the life of the option. If the stock does not pay dividends, Equation 20.2 becomes
identical to Equation 20.1.
Notice that this generalization would apply as well to European options on assets other
than stocks. Instead of using dividend income in Equation 20.2, we would let any income
paid out by the underlying asset play the role of the stock dividends. For example, European
put and call options on bonds would satisfy the same parity relationship, except that the
bond’s coupon income would replace the stock’s dividend payments in the parity formula.
Even this generalization, however, applies only to European options, as the cash flow
streams from the two portfolios represented by the two sides of Equation 20.2 will match
only if each position is held until expiration. If a call and a put may be optimally exercised
at different times before their common expiration date, then the equality of payoffs cannot
be assured, or even expected, and the portfolios will have different values.
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